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Theorem bnj1186 32280
Description: Technical lemma for bnj69 32283. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj1186.1  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )
Assertion
Ref Expression
bnj1186  |-  ( (
ph  /\  ps )  ->  E. z  e.  B  A. w  e.  B  -.  w R z )
Distinct variable groups:    w, B    ph, w, z    ps, w, z
Allowed substitution hints:    B( z)    R( z, w)

Proof of Theorem bnj1186
StepHypRef Expression
1 bnj1186.1 . . . . . 6  |-  E. z A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )
2 19.21v 1910 . . . . . . 7  |-  ( A. w ( ( ph  /\ 
ps )  ->  (
z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )  <->  ( ( ph  /\  ps )  ->  A. w ( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) ) )
32exbii 1635 . . . . . 6  |-  ( E. z A. w ( ( ph  /\  ps )  ->  ( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )  <->  E. z ( (
ph  /\  ps )  ->  A. w ( z  e.  B  /\  (
w  e.  B  ->  -.  w R z ) ) ) )
41, 3mpbi 208 . . . . 5  |-  E. z
( ( ph  /\  ps )  ->  A. w
( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )
5419.37aiv 1920 . . . 4  |-  ( (
ph  /\  ps )  ->  E. z A. w
( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) ) )
6 19.28v 1915 . . . . 5  |-  ( A. w ( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) )  <-> 
( z  e.  B  /\  A. w ( w  e.  B  ->  -.  w R z ) ) )
76exbii 1635 . . . 4  |-  ( E. z A. w ( z  e.  B  /\  ( w  e.  B  ->  -.  w R z ) )  <->  E. z
( z  e.  B  /\  A. w ( w  e.  B  ->  -.  w R z ) ) )
85, 7sylib 196 . . 3  |-  ( (
ph  /\  ps )  ->  E. z ( z  e.  B  /\  A. w ( w  e.  B  ->  -.  w R z ) ) )
9 df-ral 2797 . . . . 5  |-  ( A. w  e.  B  -.  w R z  <->  A. w
( w  e.  B  ->  -.  w R z ) )
109anbi2i 694 . . . 4  |-  ( ( z  e.  B  /\  A. w  e.  B  -.  w R z )  <->  ( z  e.  B  /\  A. w
( w  e.  B  ->  -.  w R z ) ) )
1110exbii 1635 . . 3  |-  ( E. z ( z  e.  B  /\  A. w  e.  B  -.  w R z )  <->  E. z
( z  e.  B  /\  A. w ( w  e.  B  ->  -.  w R z ) ) )
128, 11sylibr 212 . 2  |-  ( (
ph  /\  ps )  ->  E. z ( z  e.  B  /\  A. w  e.  B  -.  w R z ) )
13 df-rex 2798 . 2  |-  ( E. z  e.  B  A. w  e.  B  -.  w R z  <->  E. z
( z  e.  B  /\  A. w  e.  B  -.  w R z ) )
1412, 13sylibr 212 1  |-  ( (
ph  /\  ps )  ->  E. z  e.  B  A. w  e.  B  -.  w R z )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1368   E.wex 1587    e. wcel 1757   A.wral 2792   E.wrex 2793   class class class wbr 4376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-10 1776  ax-12 1793
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-ral 2797  df-rex 2798
This theorem is referenced by:  bnj1190  32281
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